Hydrodynamic Lyapunov modes and strong stochasticity threshold in Fermi-Pasta-Ulam models

The existence of a strong stochasticity threshold (SST) has been detected in many Hamiltonian lattice systems, including the Fermi-Pasta-Ulam (FPU) model, which is characterized by a crossover of the system dynamics from weak to strong chaos with increasing energy density $ϵ$. Correspondingly, the relaxation time to energy equipartition and the largest Lyapunov exponent exhibit different scaling behavior in the regimes below and beyond the threshold value. In this paper, we attempt to go one step further in this direction to explore further changes in the energy density dependence of other Lyapunov exponents and of hydrodynamic Lyapunov modes (HLMs). In particular, we find that for the FPU-$\beta$ and FPU-$\alpha \beta$ models the scalings of the energy density dependence of all Lyapunov exponents experience a similar change at the SST as that of the largest Lyapunov exponent. In addition, the threshold values of the crossover of all Lyapunov exponents are nearly identical. These facts lend support to the point of view that the crossover in the system dynamics at the SST manifests a global change in the geometric structure of phase space. They also partially answer the question of why the simple assumption that the ambient manifold representing the system dynamics is quasi-isotropic works quite well in the analytical calculation of the largest Lyapunov exponent. Furthermore, the FPU-$\beta$ model is used as an example to show that HLMs exist in Hamiltonian lattice models with continuous symmetries. Some measures are defined to indicate the significance of HLMs. Numerical simulations demonstrate that there is a smooth transition in the energy density dependence of these variables corresponding to the crossover in Lyapunov exponents at the SST. In particular, our numerical results indicate that strong chaos is essential for the appearance of HLMs and those modes become more significant with increasing degree of chaoticity.

Figure 1

Lyapunov spectrum of the FPU-$\beta$ model with $\mu =1.0$ and $ϵ=10$. Due to the Hamiltonian nature of the system the Lyapunov spectrum has the symmetry ${\lambda }^{\left(\alpha \right)}=-{\lambda }^{(2L-1-\alpha )}$. As can be seen from the inset, the system has four zero-value Lyapunov exponents. The system size used here is $L=128$.

Figure 2

(Color online) The largest Lyapunov exponent ${\lambda }^{\left(1\right)}$ vs energy density $ϵ$ for the FPU-$\beta$ model. The $ϵ$ dependence of ${\lambda }^{\left(1\right)}$ is fitted to a power law with the exponents $2.0$ and $0.25$ in the low- and high-energy regimes, respectively. The threshold value of the crossover between the two regimes is ${ϵ}_c\simeq 0.2$.

Figure 3

(Color online) $ϵ$ dependence of typical Lyapunov exponents in the positive branch of the Lyapunov spectrum for the FPU-$\beta$ model. Here, $\alpha$ takes values from $1$ to $121$ with the increment $10$ (from top to bottom). The density of the Kolmogorov-Sinai entropy ${\rho }_h\equiv h_{KS}∕N$ is plotted as a dashed line. Obviously, all the Lyapunov exponents roughly follow the same trend as ${\lambda }^{\left(1\right)}$.

Figure 4

(Color online) Normalized Lyapunov exponents ${\lambda }^{\left(\alpha \right)}\left(ϵ\right)∕{\lambda }^{\left(\alpha \right)}\left({ϵ}_0\right)$ vs energy density $ϵ$. Here the more or less arbitrary value ${ϵ}_0=0.9$ was chosen. Note that all the data for different Lyapunov exponents collapse on one master curve.

Figure 5

(Color online) Lyapunov exponents normalized by the entropy density ${\rho }_h$ vs energy density $ϵ$ for the FPU-$\beta$ model.

Figure 6

Snapshots of two prototypical Lyapunov vectors for the FPU-$\beta$ model with $ϵ=10$. The Lyapunov spectrum is shown in Fig. 1. The LV (with $\alpha =1$) associated with the largest Lyapunov exponent is highly localized while the LV (with $\alpha =124$) corresponding to a near-zero Lyapunov exponent is spatially extended.

Figure 7

(Color online) Static LV structure factor $S_u^{\left(\alpha \alpha \right)}\left(k\right)$ for a LV with $\alpha =50$. Here, $k_{max}$ is defined as the wave number of the highest peak in the spectrum $S_u^{\left(\alpha \alpha \right)}\left(k\right)$.

Figure 8

(Color online) (a) Contour plot of static LV structure factors for the FPU-$\beta$ model with $ϵ=10$. (b) Variation of $k_{max}$ with the index $\alpha$ of the LV. The Lyapunov vectors with $\alpha \approx 128$ are dominated by components with small wave numbers comparable to $2\pi ∕L$, which is the smallest nontrivial wave number permitted by the periodic boundary conditions used. These facts together imply the existence of hydrodynamic Lyapunov modes in this system.

Figure 9

$S\left(k_{max}\right)$ vs $\alpha$ for the FPU-$\beta$ model with $ϵ=10$. $S_{max}\left(ϵ\right)$, $S_{min}\left(ϵ\right)$, and ${\alpha }_S\left(ϵ\right)$ are defined to measure the significance of HLMs.

Figure 10

Spectral entropy $H_S\equiv -\sum S\left(k\right)\ln S\left(k\right)$ vs $\alpha$ for the FPU-$\beta$ model with $ϵ=10$. Note the definitions of the quantities $H_{max}\left(ϵ\right)$, $H_{min}\left(ϵ\right)$, and ${\alpha }_H\left(ϵ\right)$.

Figure 11

(Color online) (a) $S\left(k_{max}\right)$, (b) $k_{max}$, and (c) $H_S$ against the variation of the energy density $ϵ$ for Lyapunov vectors corresponding to the Lyapunov exponents shown in Fig. 3 for the FPU-$\beta$ model. The $ϵ$ dependence of all the quantities exhibit smooth transitions in a regime around the critical value ${ϵ}_c\simeq 0.2$ for the crossover in the Lyapunov spectrum.

Figure 12

(Color online) $S_{max}\left(ϵ\right)$ vs $ϵ$ for the FPU-$\beta$ model. For $ϵ<{ϵ}_c$, the quantity $S_{max}\left(ϵ\right)$ is nearly constant. For $ϵ>{ϵ}_c$, however, $S_{max}\left(ϵ\right)$ increases gradually with $ϵ$. These facts demonstrate that the significance of hydrodynamic Lyapunov modes grows as the energy density increases. Here, the threshold energy density ${ϵ}_c\simeq 0.2$ for the transition in the Lyapunov spectrum is shown as a dashed line.

Figure 13

(Color online) $H_{min}$ vs $ϵ$ for the FPU-$\beta$ model. The quantity $H_{min}$ is nearly constant for $ϵ<{ϵ}_c$ while it decreases gradually with increasing $ϵ$ for $ϵ>{ϵ}_c$.

Figure 14

(Color online) (a) $S_{max}∕S_{min}$ and (b) $H_{max}-H_{min}$ vs $ϵ$ for the FPU-$\beta$ model. In both quantities a change in the $ϵ$ dependence is observed as one crosses the threshold value ${ϵ}_c\simeq 0.2$.

Figure 15

(Color online) The normalized indices ${\alpha }_S∕L$ and ${\alpha }_H∕L$ are plotted against the energy density $ϵ$. Note that they are relatively close to $1.0$ for $ϵ>{ϵ}_c$. For the purpose of comparison, the quantities ${\alpha }_{\sigma d}$ and ${\alpha }_{\sigma }$ are also presented here. Their definitions are given in Figs. 27, 28, respectively.

Figure 16

(Color online) The largest Lyapunov exponent ${\lambda }^{\left(1\right)}$ vs $ϵ$ for the FPU-$\alpha \beta$ model. Similar to the case shown in Fig. 2 for the FPU-$\beta$ model, there is a crossover in the $ϵ$ dependence of ${\lambda }^{\left(1\right)}$ at ${ϵ}_c\simeq 5.0$.

Figure 17

(Color online) Similar to Fig. 3 but for the FPU-$\alpha \beta$ model. All Lyapunov exponents follow the same tendency of variation as the largest one.

Figure 18

(Color online) Similar to Fig. 4 but for the FPU-$\alpha \beta$ model. All data from different Lyapunov exponents roughly collapse onto a master curve.

Figure 19

(Color online) Similar to Fig. 5 but for the FPU-$\alpha \beta$ model. ${\lambda }^{\left(\alpha \right)}∕{\rho }_h$ are nearly constant over the regime of $ϵ$ shown, which indicates that all the Lyapunov vectors have roughly identical tendencies of change with $ϵ$.

Figure 20

(Color online) (a) $S\left(k_{max}\right)$, (b) $k_{max}$, and (c) $H_S$ against the variation of the energy density $ϵ$ for several Lyapunov vectors corresponding to the Lyapunov exponents shown in Fig. 17. The $ϵ$ dependence of all the quantities exhibits smooth transitions in a regime around the critical value ${ϵ}_c\simeq 5.0$ of the crossover in the Lyapunov spectrum.

Figure 21

(Color online) (a) $S_{max}$ and (b) $H_{min}$ vs $ϵ$ for the FPU-$\alpha \beta$ model. Both quantities have different behaviors in the regimes below and beyond the threshold value ${ϵ}_c\simeq 5$.

Figure 22

(Color online) Variation of the indices ${\alpha }_S$ and ${\alpha }_H$ with the energy density $ϵ$. Note that they are close to $1.0$ for $ϵ>{ϵ}_c$, which means that the corresponding Lyapunov exponents are close to zero.

Figure 23

(Color online) System size dependence of the Lyapunov spectrum of the FPU-$\beta$ model. The collapse of the data to a single master curve suggests that the finite-size effect on the Lyapunov spectrum can be neglected.

Figure 24

(Color online) System size dependence of the peak wave number $k_{max}$ of LVs of the FPU-$\beta$ model. A comparison between data sets with different system sizes $L$ indicates that the apparent condensation of $k_{max}$ at $2\pi ∕L$ for the LVs with $\alpha \approx L$, which can be observed in Fig. 8, is an effect of a finite system size.

Figure 25

(Color online) System size dependence of $S\left(k_{max}\right)$. The significant changes in $S\left(k_{max}\right)$ occur in the regime where $k_{max}$ changes with the system size $L$.

Figure 26

Numerical simulations show that the value of the normalized index ${\alpha }_S∕L$ (see Fig. 9) is nearly constant, irrespective of the variation in system size $L$.

Figure 27

(Color online) The fluctuations of the finite-time Lyapunov exponents are measured by the quantity $\sigma \mathbf{\left(}{\lambda }^{\left(\alpha \right)}\right(\tau \left)\mathbf{\right)}∕({\lambda }^{(\alpha +1)}-{\lambda }^{\left(\alpha \right)})$, where $\sigma \mathbf{\left(}{\lambda }^{\left(\alpha \right)}\right(\tau \left)\mathbf{\right)}$ is the standard deviation of the finite-time Lyapunov exponents with the index $\alpha$. The quantity ${\alpha }_{\sigma d}$ is defined as the index corresponding to the minimal value of the quantity $\sigma \mathbf{\left(}{\lambda }^{\left(\alpha \right)}\right(\tau \left)\mathbf{\right)}∕({\lambda }^{(\alpha +1)}-{\lambda }^{\left(\alpha \right)})$ in the regime ${\lambda }^{\left(\alpha \right)}\approx 0$.

Figure 28

(Color online) Standard deviation of the finite-time Lyapunov exponents $\sigma \mathbf{\left(}{\lambda }^{\left(\alpha \right)}\right(\tau \left)\mathbf{\right)}$ vs the index $\alpha$ for (a) $ϵ=10$ and (b) several cases with different energy densities $ϵ$. The quantity ${\alpha }_{\sigma }$ is defined as the index corresponding to the minimal value of the quantity $\sigma \mathbf{\left(}{\lambda }^{\left(\alpha \right)}\right(\tau \left)\mathbf{\right)}$ in the regime ${\lambda }^{\left(\alpha \right)}\approx 0$.